There should be no problem wearing trail shoes on the Cleveland Way,but a pair of Gortex or similar socks might be worth carrying although as it is a multi day walk you will be carrying spare socks anyway so you could soon change if you go over the tops in a bog.The difference lightweight footwear,shoes or very light boots,make on even a day walk is quite remarkable and I was converted years ago after doing the Dales Way in a pair of fell running shoes.Go for it and enjoy the walk